[QUE/CM-04016]

The ends \(A\) and \(B\) of a thin uniform rod of mass \(m\) and length \(2a\) can slide freely, \(A\) along a smooth horizontal wire \(OX\) and \(B\) along a smooth vertical wire \(OZ\), with  \(OZ\) pointing upwards. The wire frame \(OXZ\) is made to rotate with constant angular velocity \(\Omega\) about  \(OZ\).
Show that if \(B\) is above \(O\) and the angle \(OBA\) is \(\theta\), then \[H= \frac{2}{3} ma^2(\dot{\theta}^2 -\Omega^2 \sin^2\theta) +mga \cos\theta\] and that \(H\) is a constant of motion. Is \(H\) equal to total energy? What is the Hamiltonian if, instead of being rotated about \(OZ\), the frame \(OXZ\) is  made to rotate with constant angular velocity \(\omega\) about the horizontal axis \(OX\)? Is it conserved?The ends \(A\) and \(B\) of a thin uniform rod of mass \(m\) and length \(2a\) can slide freely, \(A\) along a smooth horizontal wire \(OX\) and \(B\) along a smooth vertical wire \(OZ\), with  \(OZ\) pointing upwards. The wire frame \(OXZ\) is made to rotate with constant angular velocity \(\Omega\) about \(OZ\).